Soap Films on Knots

Knotted wires support many interesting soap films.
It has recently been proved that every knotted wire
supports a soap film that does not touch the entire
wire. This goes against the usual mathematical
principle that a boundary has no boundary itself.
Also, there is a theorem that every smooth embedded
closed curve bounds a minimal oriented nonsingular manifold
(no triple lines or tetrahedral point singularities). This soap
film can be very difficult to imagine. Try it on the trefoil
below before looking at the answer.

These images were created with my
Surface Evolver
program. The Evolver datafiles are available for downloading via the link
under each image. (Some of them require Evolver features not available
until the release of version 2.10, which should be shortly.)